PureOS

PureOS is the official OS for the Librem 5.

Website: https://pureos.net/
Wiki: https://tracker.pureos.net/w/ (currently focusing on the PC/Laptop/amd64 use case)
Issue Tracker: https://tracker.pureos.net/
Repository: https://repo.pureos.net/pureos/pool/main/ (includes sources, arm64 and amd64 packages)
Package Search: https://software.pureos.net/

PureOS on the PinePhone

An unofficial/unsupported port of PureOS on the PinePhone was created by mozzwald and is maintained by clover.

Source code: http://pureos.ironrobin.net/git/clover/pinephone/ (Please report bugs here)
Build instructions: https://mozzwald.com/pp/pureos-pinephone-build-instructions.txt
Images: http://pureos.ironrobin.net/git/clover/pinephone/
username: purism
password: 123456
Info at PINE64 wiki: https://wiki.pine64.org/index.php?title=PinePhone_Software_Releases#PureOS

Basically every OS that supports the arm64 architecture can run on the Librem 5. The following are known to work (or work in progress):

Website: https://postmarketos.org
Wiki: https://wiki.postmarketos.org/wiki/
Librem 5 DevKit: https://wiki.postmarketos.org/wiki/Purism_Librem5_Devkit_(purism-librem5dev)
Status: work in progress

Website: https://ubports.com/
Documentation: https://docs.ubports.com/
Status of porting:

October 2019 Status of Ubuntu Touch on Librem 5 Smartphone
Commits in UBports for the Librem 5
(5 commits so far, last one on 2019-11-03)
UBPorts is currently prioritizing porting to the PinePhone and will adapt its porting work for the PinePhone to work on the Librem 5.